13.10 Rewrite the Eyring equation kBT eas:/Re-W/(Rn Ch k = , using the vibrational partition functions QA and QB for the reactants and Q for the transition state
The equilibrium constant K can be rewritten in terms of the partition functions of the reactants and the transition state. Equation 12.40 lets us write (13.28) QAQB There is a problem with this form of K, however: it contains a contribution to the partition function Qtot from the reaction coordinate that we have to treat more carefully. The transition state is not bound along the reaction coordinate-it falls apart either to create reactants or products-and therefore this coordinate is not a vibrational coordinate, but an extra translational coordinate, counted in addition to the center of mass translational motion of the transition state. Therefore, we break Qiot up into two pleces (13.29) where Qn is the translational partition function for motion along the reaction coordinate. Equation 3.44 gives us the translational partition function for motion in three dimensions, which is the product of the three independent partition functions for motion along X, Y, and Z. We take the cube root of that to find the partition function for motion along only one coordinate. We also divide the result by a factor of 2, because we are changing a vibrational coordi nate into a translational one. While we would integrate the bond length R for vibration from 0 to o, we integrate translational coordinates from -o to oo which introduces the factor of 2. We are left with R(13.30)
Show transcribed image text13.10 Rewrite the Eyring equation kBT eas:/Re-W/(Rn Ch k = , using the vibrational partition functions QA and QB for the reactants and Q for the transition state The equilibrium constant K can be rewritten in terms of the partition functions of the reactants and the transition state. Equation 12.40 lets us write (13.28) QAQB There is a problem with this form of K, however: it contains a contribution to the partition function Qtot from the reaction coordinate that we have to treat more carefully. The transition state is not bound along the reaction coordinate-it falls apart either to create reactants or products-and therefore this coordinate is not a vibrational coordinate, but an extra translational coordinate, counted in addition to the center of mass translational motion of the transition state. Therefore, we break Qiot up into two pleces (13.29) where Qn is the translational partition function for motion along the reaction coordinate. Equation 3.44 gives us the translational partition function for motion in three dimensions, which is the product of the three independent partition functions for motion along X, Y, and Z. We take the cube root of that to find the partition function for motion along only one coordinate. We also divide the result by a factor of 2, because we are changing a vibrational coordi nate into a translational one. While we would integrate the bond length R for vibration from 0 to o, we integrate translational coordinates from -o to oo which introduces the factor of 2. We are left with R(13.30)
